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Search: id:A128719
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| A128719 |
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Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k UUU's (triplerises) (n>=0; 0<=k<=n-2 for n>=2). A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down), and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps. |
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+0
3
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| 1, 1, 3, 6, 4, 16, 12, 8, 40, 53, 28, 16, 109, 176, 162, 64, 32, 297, 625, 633, 456, 144, 64, 836, 2084, 2677, 2024, 1216, 320, 128, 2377, 7016, 10257, 9849, 6008, 3120, 704, 256, 6869, 23218, 39378, 42222, 32930, 16928, 7776, 1536, 512, 20042, 76811, 146191
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OFFSET
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0,3
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COMMENT
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Row n has n-1 terms (n>=2). Row sums yield A002212. T(n,0)=A128720(n). Sum(k*T(n,k),k=0..n-2)=A128721(n) for n>=2).
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REFERENCES
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E. Deutsch, E. Munarini, and S. Rinaldi, Skew Dyck paths (in preparation).
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FORMULA
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G.f.=G=G(t,z) satisfies z(t+z-tz)G^2-(1-z-z^2+tz^2)G+1-tz=0.
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EXAMPLE
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T(3,1)=4 because we have UUUDDD, UUUDLD, UUUDDL, and UUUDLL.
Triangle starts
1;
1;
3;
6,4;
16,12,8;
40,53,28,16;
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MAPLE
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eq:=z*(t+z-t*z)*G^2-(1-z-z^2+t*z^2)*G+1-t*z=0: G:=RootOf(eq, G): Gser:=simplify(series(G, z=0, 14)): for n from 0 to 12 do P[n]:=sort(coeff(Gser, z, n)) od: 1; 1; for n from 2 to 11 do seq(coeff(P[n], t, j), j=0..n-2) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A002212, A128720, A128721.
Adjacent sequences: A128716 A128717 A128718 this_sequence A128720 A128721 A128722
Sequence in context: A098383 A067979 A091808 this_sequence A009782 A016615 A135097
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 30 2007
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